# 1.4: Combinations

**At Grade**Created by: Bruce DeWItt

### Learning Objectives

- Know the definition of a combination
- Be able to calculate the number of combinations using the combinations formula and with technology

We just looked at situations in which order matters. What if order does not matter? Suppose you have a younger brother or sister and your family goes out to a restaurant. There is a children's menu with activities at the restaurant that all the kids get. The owner of the restaurant has decided that each child will receive two different colored crayons to use on their menu. The restaurant happens to carry five colors of crayons: orange, yellow, blue, green, and red.

This is a situation in which the order that the child gets their two color crayons does not matter. If you gave a child a red crayon and then a blue crayon, it would be the same as if you gave the child a blue crayon followed by a red crayon. As with permutations, the first question to ask is **"Does Order Matter?"**. When the order does not matter, you are dealing with a situation that involves **combinations**.

#### Example 1

Consider the color crayon problem in the previous situation. Make a list showing all of the different color crayon combinations that might occur. Be organized so as not to repeat any combinations.

#### Solution

To be organized, use the letters O, Y, B, G, and R to represent the five colors (Orange, Yellow, Blue, Green, and Red). Alphabetizing the list to insure that we don't skip any combinations gives us BG, BO, BR, BY, GO, GR, GY, OR, OY, RY Notice that while we have BG, we don't have GB as that would be a repeat. It appears that there are 10 combinations possible.

As with the Fundamental Counting Principle, we now must ask the question "How can we find the solution quickly?" Making a list works nice, but it could get a bit messy if the restaurant had 24 colors to choose from instead of 5 because our list would get very long. Out of curiosity, you may have tried \begin{align*}_5 P _2\end{align*}. However, when you work this out, you find that this gives us a result of 20 instead of 10. We must modify this formula for situations involving combinations.

Shown below is the formula for finding how many combinations are possible when order * does not* matter.

As with the permutation formula, the 'n' stands for the number of objects available and the 'r' stands for the number of objects that will be selected.

#### Example 2

Consider the color crayon problem once again. Use the formula to find out the number of different color crayon combinations that are possible.

#### Solution

In our problem, 'n' is equal to 5 and 'r' is equal to 2. Our calculation would be \begin{align*}_5 C _2=\frac{5!}{2!(5-2)!}=\frac{5!}{2!\cdot3!}=\frac{120}{2\cdot6}=\frac{120}{12}=10\end{align*}. Be very careful that you find the result for the denominator before you divide! Find the \begin{align*}_n C _r\end{align*} command on your calculator and verify that \begin{align*}_5 C _2\end{align*} is indeed equal to 10.

#### Example 3

Suppose that there are 12 employees in an office. The boss needs to select 4 of the employees to go on a business trip to California. In how many ways can she do this?

#### Solution

We first ask whether the order that the employees are selected matters. In this case, the answer is no because either you will be going on the trip or you won't be going. Being the fourth name on the list of people who get to go is just as good as being the first name on the list. We have 12 people to select from and we will be selecting 4 or \begin{align*}_{12} C _4=\frac{12!}{4!\cdot(12-4)!}=495\end{align*}. There are 495 possible combination of groups of 4 that might be selected to go on the trip to California.

### Problem Set 1.4

#### Exercises

1) Use the formula for combinations to find the value of each expression. Use a calculator to verify each answer.

a) \begin{align*}_5 C _5\end{align*}

b) \begin{align*}_6 C _4\end{align*}

c) \begin{align*}_3 C _0\end{align*}

d) \begin{align*}_7 C _3\end{align*}

2) In how many ways can 3 cards be selected from a standard deck of 52 cards?

3) In how many ways can three bracelets be selected from a box of ten bracelets?

4) In how many ways can a student select five questions to answer from an exam containing nine questions?

5) In how many ways can a student select five questions to answer from an exam containing nine questions if the student is required to answer the first and the last question?

6) The general manager of a fast-food restaurant chain must select 6 restaurants from 11 for a promotional program. In how many different possible ways can this selection be done?

7) There are 7 women and 5 men in a department. In how many ways can a committee of 4 people be selected?

8) For a fundraiser, a travel agency has donated 5 free vacations to Mexico as grand prizes in a raffle. Suppose that 220 people paid for raffle tickets. In how many different ways can the vacation winners be selected?

9) A high school choir has 27 female and 19 male members. Two students will be selected from the choir to represent the school in the All-State Choir.

a) In how many ways can the director select two students if she decides both students will be female?

b) In how many ways can the director select two students if she decides both students will be male?

c) In how many ways can the director select two students?

d) Using your answers from a), b) and c), determine how many ways the choir director can select two students such that one student will be a male and one student will be a female?

#### Review Exercises

10) In how many ways can the team captain of a kickball team arrange the kicking order for the 7 players on the team?

11) An electronic car door lock has five buttons on it and each button has a different digit. Suppose the combination to unlock the door is 4 digits long.

a) How many different combinations are possible if digits can be repeated?

b) How many different combinations are possible if digits cannot be repeated?

12) Give the sample space for the different results that may occur if a coin is flipped twice.

13) Decide whether each situation involves permutations.

a) A teacher must pick two students from a class of 30 to put their answers on the board for problem #11 from last night's homework.

b) In order to be allowed outside to play in the rain, a 5-year old must put on socks, shoes, and boots.

c) A student has a strict bedtime of 11 pm. They need two hours to finish writing a paper, one hour for a math assignment, and two hours for a science experiment. It is 6 pm right now.